Change rates and prevalence of a dichotomous variable: simulations and applications.

Brinks R, Landwehr S - PLoS ONE (2015)

Bottom Line:
The transitions between the states are described by change rates, which depend on calendar time and on age.We develop a partial differential equation (PDE) that simplifies the use of the three-state model.In two further applications, the PDE may provide insights into smoking behaviour of males in Germany and the knowledge about the ovulatory cycle in Egyptian women.

ABSTRACTA common modelling approach in public health and epidemiology divides the population under study into compartments containing persons that share the same status. Here we consider a three-state model with the compartments: A, B and Dead. States A and B may be the states of any dichotomous variable, for example, Healthy and Ill, respectively. The transitions between the states are described by change rates, which depend on calendar time and on age. So far, a rigorous mathematical calculation of the prevalence of property B has been difficult, which has limited the use of the model in epidemiology and public health. We develop a partial differential equation (PDE) that simplifies the use of the three-state model. To demonstrate the validity of the PDE, it is applied to two simulation studies, one about a hypothetical chronic disease and one about dementia in Germany. In two further applications, the PDE may provide insights into smoking behaviour of males in Germany and the knowledge about the ovulatory cycle in Egyptian women.

pone.0118955.g008: Prevalence in a birth cohort.Age course of the prevalence of knowing about the OC in the birth cohort of Egyptian women aged 20 in 1988.

Mentions:
Fig. 7 shows the prevalence of knowing the period of highest fertility during the menstruation cycle (state B in our model). From the contour lines it becomes obvious that the prevalence peaks at some age and decreases afterwards. Take, for example, the birth cohort of women aged 20 years in 1988, i.e., all those born in 1968. The prevalence of state B is about 12% in that group in 1988. In the year 2000, the persons in the cohort are 32 years of age, and the prevalence is peaking at more than 26%. Some time later, the prevalence in this cohort decreases again. The age course of the prevalence of knowing the OC in this birth cohort is illustrated in Fig. 8. The decrease after the age of 32 implies that those women who once knew about the nature of the OC, start to forget about it. All birth cohorts in the study have this pattern of learning until a certain age (mostly around 30–40) and forgetting afterwards. This justifies the following approach: If the partial derivative (∂ / ∂t + ∂ / ∂a) p is positive, i.e. the prevalence in the birth cohort increases, we solely attribute the increase to i; if the partial derivative is negative (the prevalence in the birth cohort decreases), we attribute the decrease solely to r.

pone.0118955.g008: Prevalence in a birth cohort.Age course of the prevalence of knowing about the OC in the birth cohort of Egyptian women aged 20 in 1988.

Mentions:
Fig. 7 shows the prevalence of knowing the period of highest fertility during the menstruation cycle (state B in our model). From the contour lines it becomes obvious that the prevalence peaks at some age and decreases afterwards. Take, for example, the birth cohort of women aged 20 years in 1988, i.e., all those born in 1968. The prevalence of state B is about 12% in that group in 1988. In the year 2000, the persons in the cohort are 32 years of age, and the prevalence is peaking at more than 26%. Some time later, the prevalence in this cohort decreases again. The age course of the prevalence of knowing the OC in this birth cohort is illustrated in Fig. 8. The decrease after the age of 32 implies that those women who once knew about the nature of the OC, start to forget about it. All birth cohorts in the study have this pattern of learning until a certain age (mostly around 30–40) and forgetting afterwards. This justifies the following approach: If the partial derivative (∂ / ∂t + ∂ / ∂a) p is positive, i.e. the prevalence in the birth cohort increases, we solely attribute the increase to i; if the partial derivative is negative (the prevalence in the birth cohort decreases), we attribute the decrease solely to r.

Bottom Line:
The transitions between the states are described by change rates, which depend on calendar time and on age.We develop a partial differential equation (PDE) that simplifies the use of the three-state model.In two further applications, the PDE may provide insights into smoking behaviour of males in Germany and the knowledge about the ovulatory cycle in Egyptian women.

ABSTRACTA common modelling approach in public health and epidemiology divides the population under study into compartments containing persons that share the same status. Here we consider a three-state model with the compartments: A, B and Dead. States A and B may be the states of any dichotomous variable, for example, Healthy and Ill, respectively. The transitions between the states are described by change rates, which depend on calendar time and on age. So far, a rigorous mathematical calculation of the prevalence of property B has been difficult, which has limited the use of the model in epidemiology and public health. We develop a partial differential equation (PDE) that simplifies the use of the three-state model. To demonstrate the validity of the PDE, it is applied to two simulation studies, one about a hypothetical chronic disease and one about dementia in Germany. In two further applications, the PDE may provide insights into smoking behaviour of males in Germany and the knowledge about the ovulatory cycle in Egyptian women.